Heider and coevolutionary balance: From discrete to continuous phase transition

Structural balance in social complex networks has been modeled with two types of triplet interactions. First is the interaction that only considers the dynamic role for links or relationships (Heider balance), and second is the interaction that considers both individual opinions (nodes) and relationships in network dynamics (coevolutionary balance). The question is, as the temperature varies, which is a measure of the average irrationality of individuals in a society, how structural balance can be created or destroyed by each of these triplet interactions. We use statistical mechanics methods and observe through analytical calculation and numerical simulation that unlike the Heider balance triplet interaction which has a discrete phase transition, the coevolutionary balance has a continuous phase transition. The critical temperature of the presented model changes with the root square of the network size, which is a linear dependence in the thermal Heider balance.

Figure 1

Balanced and unbalanced configurations of the Heider balance theory and the proposed model. Empty circles have value $+1$ and filled circles have $-1$. Solid (dashed) lines are friendly (unfriendly) links and have value $+1$ ($-1$).

Figure 2

Balanced and unbalanced configuration of the proposed model. Empty (full) circles have the value of $+1$ ($-1$). Solid (dashed) lines are friendly (unfriendly) links and have value $+1$ ($-1$). (a) Two kinds of consensus are observed in our model. In each cluster, one node-link-node triplet is shown as representing the total triplets of that cluster. (b) In the bipolar state, all relations between the two clusters are unfriendly (disagreement). The shapes inside each cluster (square and pentagonal) refer only to the type of agreement within each cluster.

Figure 3

The graphical solutions for (a) coevolutionary balance and (b) thermal Heider balance model above and below their critical temperatures ($T_c$ and $T_c^{\prime }$). The parameter $q$ in our model is the node-link correlation, but in the thermal Heider balance model [22] it is the correlation between two links with a common edge. The number of nodes is 128.

Figure 4

(a),(b) Solution of Eq. (9) and the energy vs temperature for the presented model. (c),(d) Solution of the equation of state and energy vs temperature for the thermal Heider balance [22]. The number of nodes is 128.

Figure 5

Comparison of our theory with Monte Carlo simulations (triangles). The initial states for our simulation are agreement types I and II. (a) Mean of node-link correlation ($\langle {\sigma }_{ij}{s}_j\rangle$) vs temperature. (b) Mean of energy ($\langle s_i{\sigma }_{ij}{s}_j\rangle$) vs temperature.

Figure 6

Finite-size effect for the presented model in agreement type I and II initial states (Fig. 1). The deviation of theory and simulation will be smaller in the larger size.